Logarithmic approximants: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-02-26 12:59:53 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-02-26 13:03:00 UTC</tt>.

: The original revision id was <tt>~~542276828~~</tt>.

: The original revision id was <tt>542277300</tt>.

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which is about 1.4 semitones short of three octaves.

The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus __3/2__ is the perfect fifth. This can also be expressed by an explicit function: if bim(q) = (q-1)/(q+1), then ~~__q__~~ = bim(q). The inverse function can be written mib(q) = (1+q)/(1-q).

The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus __3/2__ is the perfect fifth. This can also be expressed by an explicit function: if bim(r) = (r-1)/(r+1), then __r__ = bim(r). The inverse function can be written mib(v) = (1+v)/(1-v).

Three types of approximants are described here:

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which is about 1.4 semitones short of three octaves.

The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus 3/2 is the perfect fifth. This can also be expressed by an explicit function: if bim(q) = (q-1)/(q+1), then q = bim(q). The inverse function can be written mib(q) = (1+q)/(1-q).

The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus 3/2 is the perfect fifth. This can also be expressed by an explicit function: if bim(r) = (r-1)/(r+1), then r = bim(r). The inverse function can be written mib(v) = (1+v)/(1-v).

Three types of approximants are described here:

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-02-26 12:59:53 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-02-26 13:03:00 UTC</tt>.<br>
-: The original revision id was <tt>542276828</tt>.<br>
+: The original revision id was <tt>542277300</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 32: / Line 32: @@
 which is about 1.4 semitones short of three octaves.
-The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus __3/2__ is the perfect fifth. This can also be expressed by an explicit function: if bim(q) = (q-1)/(q+1), then __q__ = bim(q). The inverse function can be written mib(q) = (1+q)/(1-q).
+The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus __3/2__ is the perfect fifth. This can also be expressed by an explicit function: if bim(r) = (r-1)/(r+1), then __r__ = bim(r). The inverse function can be written mib(v) = (1+v)/(1-v).
 Three types of approximants are described here:
@@ Line 511: / Line 511: @@
 which is about 1.4 semitones short of three octaves.&lt;br /&gt;
 &lt;br /&gt;
-The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus &lt;u&gt;3/2&lt;/u&gt; is the perfect fifth. This can also be expressed by an explicit function: if bim(q) = (q-1)/(q+1), then &lt;u&gt;q&lt;/u&gt; = bim(q). The inverse function can be written mib(q) = (1+q)/(1-q).&lt;br /&gt;
+The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus &lt;u&gt;3/2&lt;/u&gt; is the perfect fifth. This can also be expressed by an explicit function: if bim(r) = (r-1)/(r+1), then &lt;u&gt;r&lt;/u&gt; = bim(r). The inverse function can be written mib(v) = (1+v)/(1-v).&lt;br /&gt;
 &lt;br /&gt;
 Three types of approximants are described here:&lt;br /&gt;